Abstract

I study the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the nonlinear partial differential equations with constant Gaussian curvature .

1. Introduction

In recent decades, a class of transformations having their origin in the work by Bäcklund in the late nineteenth century has provided a basis for remarkable advances in the study of nonlinear partial differential equations (NLPDEs) [1]. The importance of Bäcklund transformations (BTs) and their generalizations is basically twofold. Thus, on one hand, invariance under a BT may be used to generate an infinite sequence of solutions for certain NLPDEs by purely algebraic superposition principles. On the other hand, BTs may also be used to link certain NLPDEs (particularly nonlinear evolution equations (NLEEs) modelling nonlinear waves) to canonical forms whose properties are well known [2, 3]. Nonlinear wave phenomena have attracted the attention of physicists for a long time. Investigation of a certain kind of NLPDEs has made great progress in the last decades. These equations have a wide range of physical applications and share several remarkable properties [4–6]: (i) the initial value problem can be solved exactly in terms of linear procedures, the so-called “inverse scattering method (ISM);” (ii) they have an infinite number of “conservation laws;” (iii) they have “BTs;” (iv) they describe pseudo-spherical surfaces (pss), and hence one may interpret the other properties (i)–(iii) from a geometrical point of view; (v) they are completely integrable [1, 3]. This geometrical interpretation is a natural generalization of a classical example given by Chern and Tenenblat [2] who introduced the notion of a differential equation (DE) for a function that describes a pss, and they obtained a classification for such equations of type (). These results provide a systematic procedure to obtain a linear eigenvalue problem associated to any NLPDE of this type [7].

Sasaki [6] gave a geometrical interpretation for inverse scattering problem (ISP), considered by Ablowitz et al. [4], in terms of pss. Based on this interpretation, one may consider the following definition.

Let be a two-dimensional differentiable manifold with coordinates (). A DE for a real function describes a pss if it is a necessary and sufficient condition for the existence of differentiable functions:
depending on and its derivatives such that the one-forms
satisfy the structure equations of a pss, that is,

This structure was considered for the first time by Chern and Tenenblat [2], motivated by Sasaki’s observation [6] that the equations which are the necessary and sufficient condition for the integrability of a linear problem of Ablowitz, Kaup, Newell, Segur-(AKNS-) type [4, 7–12] do describe pss. Its importance, in the present context, arises from the fact that the connection between pss and integrability of DEs goes well beyond the AKNS framework, as will be explained in Section 2. A DE for a real-valued function is kinematically integrable if it is the integrability condition of a one-parameter family of linear problems [13–20]:
in which and are -valued functions of , and ( and its derivatives) up to a finite order. Thus, an equation is kinematically integrable if it is equivalent to the zero curvature condition:
where , for each (spectral parameter or eigenvalue). In addition, a DE will be said to strictly kinematically integrable if it’s kinematically integrable and diagonal entries of the matrix introduced above are and .

The main aim of this paper is to use the geometric properties and differentiable functions in the construction of BTs for some NLEEs which describe pss.

The paper is organized as follows. In Section 2 we summarize the AKNS formulation of the ISM using the language of exterior differential forms; this language is very useful for geometry. The correspondence between NLEEs and their families of pss is established in Section 3. In Section 4 we find the BTs for some NLEEs (Liouville, Burgers, and sinh-Gordon equations, a third-order evolution equation (TOEE), a modified Korteweg-de Vries (mKdV) equation, and both families of equations I and II) which describe pss. Finally, we give some conclusions in Section 5.

2. The AKNS System for Some NLEEs

The ISM was first devised for the Korteweg-de Vries (KdV) equation [5]. Later, it was extended by Zakharov and Shabat [21] to a scattering problem for the nonlinear Schrödinger equation (NLSE) and subsequently generalized by Ablowitz et al. [4] to include a variety of NLEEs. The AKNS method consists of the following steps: (i) set up an appropriate, linear scattering (eigenvalue) problem in the “space” variable in which the solution of the NLEE plays the role of the potential; (ii) choose the “time” dependence of the eigenfunctions in such a way that the eigenvalues remain invariant as the potential evolves according to the NLEEs; (iii) solve the direct scattering problem at the initial “time” and determine the “time” dependence of the scattering data; (iv) do the ISP at later “times,” namely, reconstruct the potential from the scattering data. In this section, we concentrate on the first step of the AKNS method. As a consequence, each solution of the DE provides a metric on , whose Gaussian curvature is constant, equal to . Moreover, the above definition of a DE is equivalent to saying that the DE for is the integrability condition for the problem:
where denotes exterior differentiation, is a vector, and the matrix is traceless:
and consists of a one-parameter , family of one-forms in the independent variables , the dependent variable and its derivatives. Equation (1.2) has three one-forms , , and consisting of independent and dependent variables and their derivatives, such that the NLPDE is given by
which is, by construction, the original NLPDE to be solved. We illustrate here the following examples given by AKNS system.(a) Liouville’s equation:
(b) Burgers’ equation:
(c) sinh-Gordon equation:
(d) A TOEE [22]:
where , .(e) A mKdV equation:
where is a constant,
where , .(f) A family of equations I [23]:
where is a differentiable function of which satisfies , with , and are real constants, such that ,
(g) A family of equations II [23] similar to the family I, but with some signs changed:
where is a differentiable function of which satisfies , with , and are real constants, such that ,
keeping in mind that the parameter plays the role of the eigenvalue for the scattering problem in (2.1). Note that the one-form, , is not unique for a given NLPDE, for the scattering equations (2.1), (2.2), and (2.3) are form invariant under the “gauge” transformation:
where is an arbitrary matrix with determinant unity,
Integrability of (2.1) is,
requires the vanishing of the two form
It should be noted that the solution of these equations is of a very special kind. In general, (2.21) gives three different equations, which cannot be satisfied simultaneously by one-dependent variable . It has been pointed out [17, 24] that can be interpreted as a connection one form for the principle bundle on and as its curvature two form. The geometrical explanation of the is given in [6, 16].

3. The NLEEs Which Describe pss

Whenever the functions are real, Sasaki [6] gave a geometrical interpretation for the problem. Consider the one-forms defined by
where .

Let be a two-dimensional differentiable manifold parametrized by coordinates , . We consider a metric on defined by . The first two equations in (1.3) are the structure equations which determine the connection form , and the last equation in (1.3), the Gauss equation, determines that the Gaussian curvature of is −1, that is, is a pss. Moreover, an EE must be satisfied for the existence of forms (3.1) satisfying (1.3). This justifies the definition of a DE which describes a pss that we considered in the introduction.

We will restrict ourselves to the case where . More precisely, we say that a DE for describes a pss if it is a necessary and sufficient condition for the existence of functions , , , depending on and its derivatives, , such that the one-forms in (1.2), satisfy the structure equations (1.3) of a pss. It follows from this definition that for each nontrivial solution of the DE, one gets a metric defined on , whose Gaussian curvature is .

It has been known, for along time, that the sinh-Gordon (SG) equation describes a pss. In this paper, we extend the same analysis to include the Liouville, Burgers, sinh-Gordon equations, a TOEE, a mKdV equation, and both families of equations I and II.

Example 3.1. Let be a differentiable surface, parametrized by coordinates , . (a) Liouville’s equation.Consider
Then is a pss if and only if satisfies Liouville’s equation (2.4). (b) Burgers’ equation.Consider
Then is a pss if and only if satisfies the Burgers’ equation (2.6). (c) sinh-Gordon equation.Consider
Then is a pss if and only if satisfies the sinh-Gordon equation (2.8). (d) A TOEE.Consider
Then is a pss iff satisfies a TOEE (2.10). (e) A mKdV equation.Consider
Then is a pss iff satisfies a mKdV equation (2.12).(f) A family of equations I.Consider
Then is a pss if and only if satisfies the family of equations I (2.14). (g) A family of equations II.Consider
Then is a pss if and only if satisfies the family of equations II (2.16).

4. A Geometric Method Which Provides BTs

In this section, we show how the geometric properties of a pss may be applied to obtain analytic results for some NLEEs which describe pss.

The classical Bäcklund theorem originated in the study of pss, relating solutions of the SG equation. Other transformations have been found relating solutions of specific equations in [15, 17, 24, 25]. Such transformations are called BTs after the classical one. A BT which relates solutions of the same equation is called a self-Bäcklund transformation (sBT). An interesting fact which has been observed is that DEs which have sBT also admit a superposition formula. The importance of such formulas is due to the following: if is a solution of the NLEE and are solutions of the same equation obtained by the sBT, then the superposition formula provides a new solution algebraically. By this procedure one obtains the soliton solutions of a NLEE. In what follows we show that geometrical properties of pss provide a systematic method to obtain the BTs for some NLEEs which describe pss.

Proposition 4.1. Given a coframe and corresponding connection one-form on a smooth Riemannian surfaces , there exists a new coframe and new connection one-form satisfying the following:
if and only if the surface is pss. For the sake of clarity, we give a revised proof of [26].

Proof. Assume that the orthonormal dual to the coframes and possesses the same orientation. The one-forms and are connected by means of [27–31]:
It follows that satisfying (4.1) exist if and only if the Pfaffian system,
on the space of coordinates is completely integrable for , and this happens if and only if is pss. Geometrically, (4.1) and (4.3) determine geodesic coordinates on . Now, if describes pss with associated one-forms , (4.1) and (4.3) imply that the Pfaffian system,
is completely integrable for whenever is a local solution of [2, 32].

Proposition 4.2. Let be a NLEE which describe a pss with associated one-forms (1.2). Then, for each solution of , the system of equations for ,
is completely integrable. Moreover, for each solution of of and corresponding solution ,
is a closed one-form [2].

Eliminating from (4.5), by using the substitution
where
then (4.5) is reduced to the Riccati equations:
The procedure in the following is that one constructs a transformation satisfying the same equation as (4.10) with a potential where
Thus, eliminating in (4.9), (4.10) and (4.11), we have a BT to a desired NLEE. We consider the following examples [33].(a) BT for Liouville’s equation.For (2.4) we consider the functions defined by
for any solution of (2.4), the above functions satisfy (2.21). Then (4.9) becomes
If we choose and as
then and satisfy (4.13). If we eliminate in (4.13) and (4.10) with (4.14), we get the BT:
Equation (4.15) is the BT for Liouville’s equation (2.4) with , , and given in (4.12). (b) BT for Burgers’ equation.For any solution of the Burgers’ equation (2.6), the functions
The above functions satisfy (2.21). Then (4.9) becomes [27]
If we choose and as
then and satisfy (4.17). If we eliminate in (4.17) and (4.10) with (4.18), we get the BT:
where we put and . Equation (4.19) is the BT for the Burgers’ equation (2.6) with , and given in (4.16).(c) BT for sinh-Gordon equation.For (2.8) we consider the following functions of defined by [28]
for any solution of (2.8), the above functions satisfy (2.21). Then (4.9) becomes [29]
If we choose and as
then and satisfy (4.21). If we eliminate in (4.21) and (4.10) with (4.22), we get the BT:
Equation (4.23) is the BT for the sinh-Gordon equation (2.8) with , , and given in (4.20). (d) BT for a TOEE.For (2.10) we consider the functions defined by
The above functions satisfy (2.21). Then (4.9) becomes [30]
If we choose and as
then and satisfy (4.25). If we eliminate in (4.25) and (4.10) with (4.26), we get the BT:
Equation (4.27) is the BT for a TOEE (2.10) with , , and given in (4.24).(e) BT for a mKdV equation.For (2.12) we consider the following functions of defined by
for any solution of (2.12), the above functions satisfy (2.21). Then (4.9) becomes [31]
If we choose and as
then and satisfy (4.29). If we eliminate in (4.29) and (4.10) with (4.30), we get the BT:
where we put and . Equation (4.31) is the BT for an mKdV equation (2.12) with , , and given in (4.28).(f) BT for the family of equations I.For any solution of the family of equations I (2.14), the functions
The above functions satisfy (2.21). Then (4.9) becomes [34]
If we choose and as
then and satisfy (4.33). If we eliminate in (4.33) and (4.10) with (4.34), we get the BT:
Equation (4.35) is the BT for the family of equations I (2.14) with , , and given in (4.32).(g) BT for the family of equations II.For (2.16) we consider the functions of defined by
for any solution of (2.16), the above functions satisfy (2.21). Then (4.9) becomes
If we choose and as
then and satisfy (4.37). If we eliminate in (4.37) and (4.10) with (4.38), we get the BT:

Equation (4.39) is the BT for the family of equations II (2.16) with , , and given in (4.36).

We have previously discussed the relationships among the geometrical properties and the BT. There, we restrict our discussion to the NLEEs which can be reduced to the Liouville’s form of the geometrical properties such as the Burgers, the sinh-Gordon equations, a TOEE, a mKdV, and the two family of equations I and II.

5. Conclusions

We may hope to find some relationships among various soliton equations which describe pss. The latter yields directly the curvature condition (Gaussian curvature equal to −1, corresponding to pseudo-spherical surfaces). This geometrical method is considered for several NLPDEs which describe pss: Liouville, Burgers, sinh-Gordon equations, a TOEE, a mKdV, and the two families of equations I and II. We show how the geometric properties of a pss may be applied to obtain analytic results for some NLEEs which describe pss. This geometrical method allows some further generalization of the work on Bäcklund transformations given by Wadati et al. [3]. The Bäcklund transformations for all seven NLPDEs mentioned above are derived in this way.